Certified chemical accuracy for systems of a thousand atoms
In plain words
Chemists want energies correct to within 1 kcal/mol (about 0.043 eV), small enough to predict which reaction or crystal form wins, but the methods that reach this cost grows so fast that they fail beyond tens of atoms. The goal is a method that keeps this accuracy for proteins, crystals and surfaces and states its own error bar.
Precise statement
For systems of 100 to 1000 atoms (molecular crystal lattice energies, host-guest binding, molecules adsorbed on metal or oxide surfaces), devise an electronic-structure method with computational cost scaling at most as $O(N^3)$ in system size $N$ that delivers interaction or reaction energies within $1\,\mathrm{kcal}/\mathrm{mol}$ of the exact Born-Oppenheimer value with a computed, verifiable error estimate. An answer is the method with benchmark evidence against converged references, or a complexity argument showing the target is unattainable for a stated class of methods.
What would settle it
Blind predictions of a set of large-system energies, each with a stated error bar, later confirmed by converged reference calculations or calorimetric data to lie within 1 kcal/mol.